Entanglement and Decoherence inCoined Quantum Walks

Peter Knight, Viv Kendon, Ben Tregenna, Ivens Carneiro, MathieuGirerd, Meng Loo, Xibai Xu (IC) & Barry Sanders, Steve Bartlett

(Macquarie), Eugenio Roldan (Valencia) &John Sipe (Toronto)• Random walk as a quincunx• early mention of a quincunx: Caesar's De Bello Gallico, Book VII:

http://www.romansonline.com/sources/dbg/Ch07_73.asp (I’m impressedthat the Roman Empire set up its own web-site)

• Asterix: for the other side of the coin, translation to Englishhttp://www.aunet.org/~thaths/asterix/books/asterix_the_legionary.html

Barry Sanders Viv KendonBen Tregenna Steve Bartlett

UK Australia/Canada

Why are these walks interesting? Motivation:quantum interferencesrelate to algorithmic speed-up, can engineer coin decoherenceand study transition to classical, ….

Also John Sipe, Eugenio Roldan, I Carneiro, Mathieu Girerd, Meng Loo &Xibai Xu

Francis Galton & the Quincunx• Galton’s quincunx demonstrates

random walk.• Each peg is like the coin, and the

ball goes L or R.• Distribution of balls follows

normal distribution withvariance proportional to numberof rows of pegs.

• How to design a quantumquincunx to show quantumwalk?

Reminder of usual 1D classical random walk• Particle confined to motion along a regular lattice in 1D, with points

indexed by integers i (integer time)• Decide on whether to go forward or backward according to whether

an unbiased coin reveals heads (+ ) or tails (-).• The distribution P(i) is binomial and approaches a Gaussian

distribution after many steps.• The average distance from the origin increases according to

ts ∝Sqrt(time) Dependence

0

200

400

600

800

1000

1200

1

30

59

88

117

146

175

204

233

262

291

320

349

378

407

436

465

494

523

552

581

610

639

668

697

726

755

784

813

842

871

900

929

958

987

Step

(St.

Dev

.)^

2

(St.Dev.)^2

Also Julia Kempe, Dorit Aharanov, and many peoplehere have contributed to math description of walks….Julia Kempe review: Contemp Phys 2003/ quant-ph/0303081.-but realisations?-Decoherence?

Quantum coin for Quantum Quincunx• Coin replaced by quantum two-level system:

Quantum coin toss - Hadamard transformation:

Quantum Quincunx

• Head for L, Tail for R:

Superposition of headsand tails goes tosuperposition of left andright: interferences!

Chose symmetric initialstate…

Quantum superpositions with coin flip:

Conditional Translation of Position• The particle’s translation depends on the state of the

coin, and the entire quantum random walk isdeterministic (in a wave sense) over the joint particle-coin Hilbert space. Quantum walk is not a RandomWalk..

• Is it always “quantum?”-- see later

“walk this way….”

Symmetric walk

Square lattice: Hadamard coin

Square lattice: Grover coin

Decoherence: walk on line (Kendon & Tregenna

Entropy for walk on line

Entropy depends on initial conditions:approachesasymptote faster if more symmetric

Entropy for symmetric but biased coin

Entropy for asymmetric ics oscillates between a maxof 1 and a min of 0

Bouwmeester, Schleich et al- already observed quincunx in classicaloptical experiment. Described in terms of many Landau-Zener crossings

Ions: Travaglione and Milburn;Lattices: Kendon and Briegel etc

Its not random. Is it quantum?• Knight, Roldan & Sipe (quant-

ph 0305165/Opt Comm inpress) optical implementationof walk for polarization orposition “cebits”

• Bouwmeester et al alreadyimplemented in their GaltonBoard experiment

• But look at resources: cancompute entanglement interms of entropy usingSchmidt basis - 4-component=4 propagating spinwaves…entanglement!

The proposed setup

T2T2

T1

T2

T2T2 T2T2 T2

T2 T2

T2 T2T2T2

T2j=1

j=0

k=+1

k=0 k=-1

D0

D-2

D-4

D+4

D+2

dynamiclines

node

Scheme of Paternostro et al

When the walker is a single-photon state: 4th step

T2T2

T1

T2

T2T2 T2T2 T2

T2 T2

T2 T2T2T2

T2

D0

D-2

D-4

D+4

D+2

s1

-4-2024Position0.050.10.150.20.250.30.35Np-4-2024Position0.050.10.150.20.250.30.35Np classical distribution quantum distribution

-40-2002040Position0.020.040.060.080.1Np50th step

Simulating QRW with any state of light! (1)The proposed optical set-up realizes the following unitary evolutionof the state of the walker after N steps

00ˆˆˆˆˆ Φ≡Φ=Φ ====NQW0)1(j1)T(j2)T(jN)T(jN UTUU...U

NNN 22211

ˆ

0 ... αχαχαχα =Φ→=ΦNQWU

For an input coherent state, this is easily explicitly calculated:

ααχαχαχαχαχαχα

ρρααααρ

2

ˆ

...)(

ˆˆ)(

N2NN222111

NQW 0

N QW N

U2 0

P

U UP NQW

d

d

∫∫

⊗⊗⊗=

=→=+

with χi that depend on the structure of the set-up.

For an arbitrary state, in the Glauber diagonal P-representation

Introducing randomness: decoherence in QRW

Let’s suppose we introduce an additional phase shift between twosuccessive operations T2 in the proposed set-up. We shift the state of the

walker, before each T2 , by 2πl, where l is randomly chosen from aGaussian distribution centred at 1 and with standard deviation σps. In

practice, the random phase shift (RPS) is as follows:

T2

T1 T2T2 T2T2

T2

T2 l,l,l are taken from

σps

1

Equal random shift 2πl

Equal random shift 2πl

Equal random shift 2πl

α

Quantum Quincunx in a Cavity:Phys Rev A67 (2003) 042305

• A two-level atom is the coin: a conditional Stark term shifts thephase of the cavity field clockwise or counterclockwise dependingon the state of the atom.

• Sequence of atoms for repeated random phase shifts: classical walk.• The cavity field initially has a “sharp” phase distribution (NB

domain is a circle, not a line).• variance grows linearly with number of atoms: phase diffusion

Recycling the Coin: quantum walk• The quantum ‘coin’ corresponds to the two levels of the atom, and this coin can be

recycled to give a quantum walk.• The cavity evolution is interrupted by periodically spaced Hadamard

transformations, which ‘flip’ the coin: F(ϕ) operates on the atom-cavity system inbetween these Hadamard coin flips.

• It is conceivable to apply (FH)15 within the timescale of an experiment (Paris groupparameters: J M Raimond, private communication)

• . For small ϕ, the results are similar to 1D random walk, but large ϕ is also possible:random walk on a circle.

Quantum Quincunx• Use a single atom (recycle the

coin)• Use π/2 pulse to implement

quantum coin flip• Quantum phase diffusion =

quantum quincunx• Phase spreads quadratically faster• Need open cavity (Haroche)

Conditional phase shifts?• Conditional phase-shift operator• Cavity prepared in a coherent state and the atom in

either the + or – state:• The phase of the cavity coherent state undergoes a

random walk in discrete steps of ϕ.• Ideally consider the (un-normalized) phase state

( ) ( )zaaiF σϕ=ϕ +exp

( ) ±⊗α=±⊗αϕ ϕ± 2/ieF

( ) ∑∞

=

φ∝φ±⊗ϕ±φ=±⊗φϕ0

for ,n

in neF

Wigner functions, α=3, phase step 2π/6?

T=4 T=3

T=2T=1

Note fringes as well as displacements-see via quadraturemeasurements

Quadrature variances

conclusions• Quantum version of Galton’s quincunx shows differences from

classical walks: interferences, quantum spreading• Feasible: Haroche group developed new cavity allowing Hadamards

during transit of atom (Yamaguchi quant ph-02)• Quadratic enhancement of phase diffusion: quantum speed up;

quantum algorithms• Can engineer decoherence: observe transition from quantum to

classical, suppression of interferences• Funding: